This paper provides the first part of a study on the effect of magnesium on the structural phase composition and physical and mechanical properties of nanostructured aluminum–magnesium composite materials with the composition AlxMgy + 0.3 wt % C60 fullerene. Composite powders are obtained by the simultaneous mechanical activation of the initial materials in a planetary ball mill in an argon atmosphere. It is found that the powders have a complex hierarchical structure made up of 50–200 μm aggregates consisting of 5–10 μm strong high-density agglomerates, which in turn are a combination of nanoscale (30–60 nm) crystallites. It is found that the increase in magnesium concentration in the composite up to 18 wt % makes it possible to obtain crystallites with an average size of less than 30 nm during mechanical activation, while the size of aggregates is less than 50 μm. The maximum solubility of magnesium in aluminum with a crystallite size of 30–70 nm during mechanical activation is 15 wt % (17 at %). Using the differential scanning calorimetry method, it is found that nanostructured composites undergo irreversible structural phase transformations during heat treatment in a temperature range of 250–400°C: recrystallization, decomposition of the α-solid solution of magnesium in aluminum, and the formation of intermetallic β-(Al3Mg2), γ-(Al12Mg17) and carbide (Al4C3) phases. In addition, the Raman spectra contain peaks that, according to some sources, correspond to covalent compounds of aluminum with C60 fullerene—aluminum–fullerene complexes. The data that have been obtained will be used in further research to determine the parameters for the thermobaric treatment of nanocomposite powder mixtures in order to obtain and test bulk samples.